\Chapter{Theoretical background}
\label{chap:theory}
\ifpdf
    \graphicspath{{2_TheoreticalBackground/TheoreticalBackgroundFigs/}
    							{4_ResultsandDiscussion/ResultsandDiscussionFigs/}
    							{Structure/cpgsFigs/}}
\else
    \graphicspath{{2_TheoreticalBackground/TheoreticalBackgroundFigs/EPS/}
    							h{2_TheoreticalBackground/TheoreticalBackgroundFigs/}}
\fi

\section{Electromagnetics of metals}
\label{sec:EMmetals}
The plasmonic responses discussed in this work can be adequately understood in the context of classical electrodynamics based on Maxwell's equations. Such a treatment is satisfactory even for metallic structures with sizes down to the order of a few tens of \nm{\!} as a result of the very high density of free charge carriers in metals. This high carrier density ensures that the spacings between electron energy levels are small compared to thermal excitations at room temperature\footnote{typical free electron densities in metals are of the order \cmp{10^{23}}{-3}. Typical spacings, even between the conduction and valence band, are of the order of a few \unit{\!}{meV}, while spacings of levels within the bands are of course smaller still. All of these are small compared to thermal excitations at room temperature, for which $k_B T=25.6\unitspace{\rm meV}$ at \unit{25}{^\circ C}}. In a similar vein, we may also keep to the use of the macroscopic forms of the Maxwell equations, provided all structures are significantly larger than critical dimensions of the material, such as the lattice constant and electron mean free path. For structures considered in this report, we consider materials as continuous.

I begin by stating the four macroscopic Maxwell equations, in the form

\begin{eqnarray}
\label{eq:MaxwellDivD} \Div  \D    &=&  \rho_{\rm ext}\\
\label{eq:MaxwellDivB} \Div  \B    &=&  0\\
\label{eq:MaxwellCurlE} \Curl \E    &=& -\ \partiald{\B}{t}\\
\label{eq:MaxwellCurlH} \Curl \X{H} &=&  \X{J_{\rm ext}}+\partiald{\D}{t}
\end{eqnarray}

The materials of interest to us, principally noble metals along with various dielectric substrates, are non-magnetic, so that we may set the magnetization $\X{M}=0$. We also limit the discussion to media with a linear polarization response, such that $\X{P}=\epsilon_0\chi\E$. For materials showing no dispersion, this gives us 

\begin{equation}
\Drt=\epsilon_0\epsilon\Ert
\label{eq:DispersionlessD}
\end{equation}

However, in general, the response of the medium will exhibit both temporal and spatial dispersion, so
\begin{equation}
\Drt=\epsilon_0\int dt'\X{d\X{r}}'\epsilon\fn{\X{r}-\X{r}',t-t'}\E\fn{\X{r}',t'}
\label{eq:TimeDomainD}
\end{equation}
With $\epsilon\fn{\X{r}-\X{r}',t-t'}$ describing the impulse response of the material polarization. The treatment of \fref{eq:TimeDomainD} can be significantly simplified by considering the response in the Fourier domain. Since we defined the response as linear, we may, via Fourier transforms from $t$ to $\omega$ and $\X{r}$ and $\X{K}$, replace the convolutions by multiplications such that

\begin{equation}
\D\fn{\X{K},\w}=\epsilon_0\epsilon\fn{\X{K},\w}\E\fn{\X{K},\w}
\label{eq:freqDomainD}
\end{equation}

Provided that the wavelength in the material is significantly longer than the material's own characteristic length scales, such as the electron mean free path and the lattice constant, we may also simplify our treatment to a spatially local response, so that the spatial response becomes a delta function $\epsilon\fn{\X{r}-\X{r}',t-t'}\rightarrow\delta\fn{\X{r}-\X{r}'}\epsilon\fn{t-t'}$, and

\begin{equation}
\D\fn{\X{K},\w}=\epsilon\fn{\w}\E\fn{\X{K},\w}
\label{eq:DeqEpsE}
\end{equation}

In order to determine plasmonic responses of metals, we must therefore consider the functional form of the metal's dielectric response $\epsilon\fn{\omega}$. It is the fact that the dielectric response of metals varies significantly over the spectrum which prevents us from simply applying scale invariance to the Maxwell equations and designing nanometer scale optical components as smaller versions of designs proven to be effective in radio and microwave frequency ranges. As a starting point, we consider the dielectric function of a free electron gas, in which a gas of free electrons moves over a distribution of positive ion cores. The free electron gas model can adequately represent metals' dielectric functions over a wide frequency range, extending even up to UV for alkali metals. For noble metals such as those used for plasmonic structures, interband electronic transitions in the visible frequency range complicate the picture, as I will later discuss. A free electron gas driven by a harmonically varying field and damped by collision occurring with a frequency $\gamma$ may be described by the equation of motion

\begin{equation}
m\partialdd{r}{t}+m\gamma\partiald{r}{t}=-e\E_0\e^{-i\w t},
\label{eq:FreeElectronGas}
\end{equation}

where $m$ is the electrons' effective mass. A solution of \fref{eq:FreeElectronGas} is $\X{r}\fn{t}=\X{r_0}e^{-i\w t}$. Applying this to the equation of motion, we obtain

\begin{equation}
\X{r}\fn{t}=\frac{e}{m\fn{\w^2 +i\gamma\w}}\E\fn{t}
\label{eq:rFEG}
\end{equation}

Since the bulk polarization $\X{P}$ may be written as $-ne\X{r}$, with $n$ the number density of free electrons, and recalling that $\X{P}=\epsilon_0\chi\E$ and $\epsilon=1+\chi$, we may deduce that for the free electron gas,

\begin{equation}
\epsilon\fn{\w}=1-\frac{{\w_p}^2}{\w^2+i\gamma\w}
\label{eq:FreeElectronDielectricFn}
\end{equation}

where $\w_p=\sqrt{n e^2/\fn{m\epsilon_0}}$ is the so-called plasma frequency for the free electron gas. The metal remains metallic only for frequencies $\w<\w_p$, above which it acquires a dielectric nature, as observed in the ultraviolet transparency of most metals. For $\omega\gg\omega_p$, $\epsilon\rightarrow1$ in this free electron picture. However, for noble metals, the filled bands below the `free' conduction band result in a large polarization even for frequencies $\omega>\omega_p$, and the dielectric function requires modification to

\begin{equation}
\epsilon\fn{\w}=\epsilon_\infty-\frac{{\w_p}^2}{\w^2+i\gamma\w}
\label{eq:epsDrude}
\end{equation}

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=0.9\textwidth]{DrudeDielectric}%
	\caption{The curves show the real and imaginary parts of a dielectric function of the form of \fref{eq:epsDrude}, fitted for gold. Also plotted as points are the experimentally measured values used for the fit, as recorded in \cite{JohnsonAndChristy}. Although a good fit for longer wavelengths, it is clear that for $\lambda<600\unitspace{\rm nm}$, the Drude-Sommerfeld model of a free electron gas is insufficient to describe the measured response.}%
	\label{fig:Drude}%
\end{center}
\end{figure}

where $\epsilon_\infty$ is usually between 1 and 10 or so. The real and imaginary parts of the dielectric function $\epsilon\fn{\w}$ using \fref{eq:epsDrude} is displayed in \fref{fig:Drude}, fitted to experimental data for gold. It is clear that although the function fits the well for longer wavelengths, for those of around \nm{550} and below, the dielectric function of gold differs significantly from that predicted by the Drude-Sommerfeld model. The reason for this is transitions of electron between electronic bands, which can be efficiently excited in gold by light of shorter wavelengths, limiting the validity of the free-electron model. In order to account for such interband transitions, the model must be extended to include the response of bound electrons in lower-lying valence bands, as well as those in the quasi-free conduction band. This can be accomplished for a single transition frequency $\w_j$ by applying the same method again, to the equation of motion for bound electrons:

\begin{equation}
m_i\partialdd{r}{t}+m_j\gamma_j\partiald{r}{t}+\alpha_j\X{r}=-e\E_0\e^{-i\w t},
\label{eq:BoundElectron}
\end{equation}

Using the same solution as for \fref{eq:FreeElectronGas}, the contribution of the bound electron to the dielectric function is found to be of the form

\begin{equation}
\epsilon_{j}=1+\frac{A_i}{\fn{{\w_j}^2-\w^2}-i\gamma_j}.
\label{eq:boundDielectric}
\end{equation}

The constant $A_j$ is given by $A_j=n_je^2/\fn{m_j\epsilon_0}$, where $n_j$ is the density of bound electrons with resonant frequency $\w_j$, and $m_j$ their effective mass. The constants, as well as $\gamma_j$, are in general different for different $\w_j$. A number of such Lorentz oscillator terms can be added to the free electron gas dielectric function in order to account for the effects of interband transitions. A dielectric function composed of contributions from both free and bound electrons is plotted in \fref{fig:Interband}. However, although this technique can produce a reasonable dielectric function for the metal, it does not tell the whole story. For example, electrons excited by interband transitions at an energy $\hbar\w_j$ may decay entirely non-radiatively, but equally it is possible for them to relax non-radiatively before re-emitting a photon at a reduced frequency $\w<\w_j$, in a fluorescence process. The profile of the emission is unrelated to the initial excitation. Although the dielectric function accounts for the losses caused to the exciting field by interband absorption, it does not account for fluorescence from the excited electrons. Such fluorescence can be relevant to the function of plasmonic devices, and so it must be borne in mind that the entire material response cannot be encompassed by a dielectric function alone.

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=0.9\textwidth]{DrudeLorentzDielectric}%
	\caption{Real and imaginary parts of a dielectric function for gold composed of a Drude-type free electron as in \fref{eq:epsDrude} term plus Lorentz-oscillator bound terms as in \fref{eq:boundDielectric}. Also plotted are experimentally measured values for the gold dielectric constant, as recorded in \cite{JohnsonAndChristy}.}%
	\label{fig:Interband}%
\end{center}
\end{figure}

\revisit{
\begin{itemize}
	%\item \done{media: non-magnetic and linear}
	%\item \done{frequency domain treatment (easier for impulse response).}
	%\item \done{Dielectric functions of metals - strongly frequency dependent, prevents scale invariance from making sense}
	%\item \done{Drude model} \TODO{figure}
	%\item \done{Drude model plus local Lorentz oscillator interband transition(s)} \TODO{figure}
	\item Noble metal use, as a result of plasma frequencies \TODO{why?}
	%\item \done{interband transitions not modelled by using $\epsilon\fn{\w}$ - effects such as fluorescence (and resultant antenna feeding) are missed}
\end{itemize}}


















\section{Analytical solutions}
\label{sec:analytic}

\subsection{Planar interface}
\label{sec:planarInterface}
In this section, I will outline a solution of the macroscopic Maxwell equations for a planar interface between two half-spaces, which shows characteristic plasmonic confinement, broadly following the treatment in \cite{NovotnyAndHecht}. This solution is valid only for certain combinations of materials, as we will discover. To begin, consider the electromagnetic wave equation

\begin{equation}
\Curl\Curl\widetilde{\E}\fn{t}=-\mu_0\partialdd{\widetilde{\D}\fn{t}}{t}
\label{eq:WaveEqn}
\end{equation}

Via a Fourier transform into the frequency domain, noting that $\mu_0\epsilon_0=c^{-2}$, and that in frequency space $\D\fn{\w}=\epsilon\fn{\w}\E\fn{\w}$, we obtain the following

\begin{equation}
\Curl\Curl\E\fn{\w}=\frac{\w^2}{c^2}\epsilon\fn{\w}\E\fn{\w}
\label{eq:FourierWaveEqn}
\end{equation}

Since we pursue a solution in a region of space with no driving charges distribution, we may  use the identity $\Div\D=0$ to show that

\begin{equation}
\Div\E=-\frac{1}{\epsilon}\E\cdot\Grad\epsilon
\label{eq:DivE}
\end{equation}

If we assume that there is negligible variation in $\epsilon$, at least on a length scale comparable with the wavelength, then we may simplify the above to give the Helmholtz equation

\begin{equation}
\Del^2\E+{k_0}^2\epsilon\E=0
\label{eq:Helmholtz}
\end{equation}

where $k_0={\w}/{c}$ is the free-space wave vector. Considering the geometry in \fref{fig:InterfacePlasmon},  we assume a solution which propagates in the $x$ direction, and is confined to the interface in the $z$ direction, and exhibits no variation in the $y$ direction. We may consider both \TM\ and \TE\ polarized versions of the solution, in which the electric field vector is respectively either parallel or perpendicular to the $xz$ plane. 

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=60mm]{InterfacePlasmon}%
	\caption{Geometry of the interface solution. We seek a solution to the Helmholtz equation describing a wave which propagates in the $x$ direction, and is confined in the $z$ direction to the interface at $z=0$}%
	\label{fig:InterfacePlasmon}%
\end{center}
\end{figure}

The \TM\ polarized version is of the form

\begin{equation}
\E=\left(\!\!\begin{array}{c}{\rm E}_{j,x}\\0\\{\rm E}_{j,z}\end{array}\!\!\right)\exp{\left\{i\fn{k_{j,x} x + k_{j,z} z -\w t}\right\}}
\label{eq:TMSolution}
\end{equation}

where the subscript $j=1,2$ denotes the half-space in which the solution applies. Using $\Div\D=0$, we find

\begin{equation}
k_{j,x}\E_{j,x}+k_{j,z}\E_{j,z}=0
\label{eq:DivDZero}
\end{equation}

Using the standard boundary conditions, namely the continuity of the $\E$ components parallel to the interface and the $\D$ components perpendicular to it, we obtain

\begin{eqnarray}
\nobreak
\label{eq:BoundaryCond1}k_{1,x}-k_{2,x}&=&0\\
\label{eq:BoundaryCond2}\epsilon_1{\rm E}_{1,z}-\epsilon_2{\rm E}_{2,z}&=&0
\end{eqnarray}

The coupled set of equations \ref{eq:DivDZero}, \ref{eq:BoundaryCond1} and \ref{eq:BoundaryCond2} show two alternative classes of solution. The first is trivial, for $k_{j,z}=0$, which is non-propagating and hence not of interest to us. The alternative solution demands that

\begin{equation}
\epsilon_1 k_{2,z}=\epsilon_2 k_{1,z}
\label{eq:CoupledSolution}
\end{equation}

which in turn gives us the dispersion relation

\begin{equation}
{k_x}^2=\frac{\epsilon_1\epsilon_2}{\epsilon_1+\epsilon_2}\frac{\w^2}{c^2},
\label{eq:DispersionRelation}
\end{equation}

along with the associated relation

\begin{equation}
{k_{j,z}}^2=\frac{{\epsilon_j}^2}{\epsilon_1+\epsilon_2}\cdot
\label{eq:zDisp}
\end{equation}

If we assume that the dielectric function of the medium in the lower half space has negligible imaginary component,  in comparison to their real parts, we may see from \fref{eq:CoupledSolution} that in order for the solution to be bound to the interface, the wave vector components $k_{j,z}$ must be purely imaginary. For it to be a propagating solution, we require $k_x$ to be real. These conditions, together with equations \ref{eq:DispersionRelation} and \ref{eq:zDisp}, imply that
\begin{eqnarray}
\nobreak
\epsilon_1+\epsilon_2 & < & 0\\
\epsilon_1\epsilon_2 & < & 0
\label{eq:epsilonConditions}
\end{eqnarray}

That is, that one of the dielectric functions must be both negative and greater in magnitude than the other (positive) dielectric function. As discussed in the previous section, the dielectric functions of metals exhibit large negative real components in the optical frequency range, and thus we can expect to see such confined modes at the interface between a metal and a dielectric. The modes are composed of a coherent charge density oscillation in the metal, coupled to an evanescent electromagnetic wave in the dielectric, and as a result are referred to as surface plasmon polaritons. Now briefly consider the possibility of analogous \TE\ polarized modes, such that

\begin{equation}
\E=\left(\!\!\begin{array}{c}0\\{\rm E}_{j,y}\\{\rm E}_{j,z}\end{array}\!\!\right)\exp{\left\{i\fn{k_{j,x} x + k_{j,z} z -\w t}\right\}}\cdot
\label{eq:TESolution}
\end{equation} 

For the proposed \ solution, $\Div\D=0$ implies that $k_{j,z}{\rm E}_{j,z}=0$. In order to have the solution confined to the interface, we require that both $k_{j,z}$ are non-zero, and hence must conclude that for the \TE\ solution, ${\rm E}_{j,z}=0$. Similarly to the \TM\ analysis, continuity of the $y$ component of $\E$ across the boundary implies ${\rm E}_{1,y}={\rm E}_{2,y}$. From the $x$ component of the curl equation \ref{eq:MaxwellCurlE}, we note that

\begin{equation}
\partiald{\E}{z}\cdot\X{n}_{y}=i\mu_0\w{\rm H}_x\fn{t}
\label{eq:TEcurl}
\end{equation}

Another standard boundary condition for piecewise homogeneous media is the continuity of parallel components of $\X{H}$ across the boundary. This imposes the condition that $k_{1,z}\E=k_{2,z}\E$, however in order to ensure confinement of the fields to the interface, we require the two $k_{j,z}$ to be of opposite sign. Since this can only be satisfied when ${\rm E}_y=0$, there can be no \TE\ solution for surface plasmon polaritons at the planar interface.


\subsection{Plasmon resonances of particles}
\label{sec:spherical_solution}
A second simple analytical solution which gives interesting results is that of a small metal sphere. A complete theory of the scattering of light from colloidal particles was presented by Gustav Mie in his 1908 paper\cite{Mie}. However, for particles sufficiently small compared with the excitation wavelength, we may use the simple quasi-static approximation, in which the applied field is assumed to be constant across the particle volume, i.e. phase delay effects are ignored. In this limit, we seek the solution of the Laplace equation for the potential $\Del^2\Phi=0$, with the electric field given by $\E=-\Del\phi$ in an azimuthally symmetric geometry. The situation is outlined in \fref{fig:sphericalSoln}. Following the treatment in \cite{Jackson} for a constant field $\E=\E_0={\rm E}_0\X{n}_z$, we see that the solution for an azimuthally symmetric geometry is given by

\begin{equation}
\Phi\fn{r,\theta}=\sum^{\!\infty}_{\!l=0}\left[{\rm A}r^l+{\rm B_l}r^{-\fn{l+1}}\right]{\rm P}_l\fn{\cos\theta}
\label{eq:laplaceSolution}
\end{equation}

where ${\rm P}_l\fn{x}$ are the Legendre polynomials of order $l$ in $x$, and the constants ${\rm A}_l$ and ${\rm B}_l$ are distinct inside and outside of the sphere. The constants may be constrained by requiring the potential $\Phi$ to be finite at the origin (such that ${\rm B}_{l,{\rm in}}=0$), and requiring that $\Phi\rightarrow E_0 r\cos\theta$ as $r\rightarrow\infty$, such that ${\rm A}_{1,\rm out}=-{\rm E}_0$, and ${\rm A}_{l,\rm out}=0$ for $l\neq0$. We may then apply the standard boundary conditions of continuity of the of the electric field components tangential to the interface, and the perpendicular (that is, radial, in this geometry) components of the electric displacement $\D$. Following \cite{Jackson}, we learn that the latter two constraints lead to the standard result for a small dielectric sphere

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=80mm]{sphereSketch}%
\caption{Geometry for the solution of a homogeneous sphere in a uniform dielectric, subject to a uniform electrostatic field ${\rm E}_0$}%
\label{fig:sphericalSoln}%
\end{center}
\end{figure}

\begin{eqnarray}
\label{eq:phiLaplaceIn}\Phi_{in}&=&-\frac{3\epsilon_m}{\epsilon+2\epsilon_m}{\rm E}_0r\cos\theta\\
\label{eq:phiLaplaceOut}\Phi_{out}&=&{\rm E}_0r\cos\theta +\frac{\epsilon-\epsilon_m}{\epsilon+2\epsilon_m}{\rm E}_0a^3\frac{\cos\theta}{r^2}
\end{eqnarray}

The electric field can be calculated from the potentials $\Phi$ to give

\begin{eqnarray}
\label{eq:sphereEFieldIn}\E_{\rm in}&=&{\rm E}_0\frac{3\epsilon_m}{\epsilon+2\epsilon_m}\X{n}_z\\
\label{eq:sphereEFieldOut}\E_{\rm out}&=&{\rm E}_0\X{n}_z + \frac{\epsilon-\epsilon_m}{\epsilon+2\epsilon_m}\frac{a^3}{r^3}\fn{3\X{r}\fn{\X{r}\cdot\X{n}_z}-\X{n}_z}{\rm E}_0
\end{eqnarray}

The scattering term (the second term of \fref{eq:sphereEFieldOut}) can be seen as that of a dipole of moment $\mu$ located at the origin (the centre of the sphere). The dipole is induced by the applied field $\E_0$ such that $\mu=\epsilon_0\epsilon_m\alpha\E_0$, with the polarizability $\alpha$ given by

\begin{equation}
\alpha=4\pi a^3\frac{\epsilon-\epsilon_m}{\epsilon+2\epsilon_m}
\label{eq:polarizability}
\end{equation}

The polarizability exhibits a resonance for the minimum of $\left|\epsilon+2\epsilon_m\right|$. For metal particles at visible frequencies, where the real part of the dielectric function dominates, we may simplify this condition to ${\rm\Re e}\left[\epsilon\fn{\w}\right]=-2\epsilon_m$. The oscillating mode associated with this relation is the particle's dipole surface plasmon mode. We can see from \fref{eq:polarizability} the strong dependence of the resonance on the local dielectric environment: as $\epsilon_m$ increases, the resonant $\epsilon$ of the metal sphere must also increase. Recalling from \fref{sec:EMmetals} that for noble metals in the optical frequency range, $\epsilon\fn{\w}$ falls for increasing wavelength, we deduce that for increasing $\epsilon_m$ the resonance will red-shift. Indeed, this effect of the dependence of plasmon resonance on the surrounding dielectric environment has been employed for optical sensing of changes to local refractive indices \cite{Mock2003}. Of interest experimentally are the particle's scattering and absorption cross sections. The scattering cross section is obtained by dividing the total power radiated by the dipole over $4\pi$ solid angle by the intensity of the exciting plane-wave field, to give

\begin{equation}
\sigma_{scat}=\frac{k^4}{6\pi{\epsilon_0}^2}\left|\alpha\fn{\w}\right|^2
\label{eq:scatteringCS}
\end{equation}

while the absorption coefficient is given by the power dissipated in the particle, which can be calculated from the Poynting vector, derived for a harmonic field from \fref{eq:sphereEFieldIn}. The absorption coefficient can be calculated as (see \cite{Maier})

\begin{equation}
4\pi k a^3 \Im {\rm m}\left[\frac{\epsilon-\epsilon_m}{\epsilon+2\epsilon_m}\right]
\label{eq:absorptionCS}
\end{equation}

It is clear from equations \ref{eq:scatteringCS} and \ref{eq:absorptionCS} that, as expected from the polarizability \ref{eq:polarizability}, the scattering and absorption coefficients are resonantly enhanced when the condition ${\rm\Re e}\left[\epsilon\fn{\w}\right]=-2\epsilon_m$ is fulfilled. It is also instructive to note that the magnitudes of the two cross sections are strongly dependent on particle size. Since scattering cross section scales $\propto a^6$, it is difficult to pick out the resonance of small particles in a background of larger scatterers. Since the scattering cross section is directly dependent on $k=2\pi/c$ as well as indirectly through $\epsilon\fn{\w}$, the peak wavelength of the scattering cross section is also dependent on the particle size. This effect is what allows different colours of stained glass to be produced by the addition of different sizes of colloidal gold particles (although the mechanism for the colouring was not known to those originally employing the technique).

%The above treatment in the quasi-static approximation is, as already mentioned, valid only for particles significantly smaller than the wavelength. Although Mie theory is necessary for a full treatment of the scattering problem, useful insight can be gained by a polynomial expansion of the Mie-scattering coefficient $^eB_1$, as described in \cite{Wokaun}. An expansion of $^eB_1$ up to order $q^3$ gives
%
%\begin{equation}
%\P=\frac{1-1/10\fn{\epsilon+\epsil}}{}
%\label{eq:wokaunPolarizability}
%\end{equation}
%
%where $q=ka$ \cite{Wokaun}.
%
%
%This treatment takes into account the retardation of the depolarization field across the particle, and results in a modification of the polarizability to




The treatment of a spherical particle outlined above can be generalized to ellipsoidal coordinates, with the case of spheroidal\footnote{a spheroid is an ellipse in which the magnitudes of two of the semi-major axes are the same, thus it is a sphere stretched in one dimension} particles showing two resonances, corresponding to oscillations along or perpendicular to the particle's abnormal axis. In general, structures with reduced symmetry, such as bars or triangles, have multiple spectral resonances associated with different spatial modes of the field. Structures formed by superposing multiple simpler modes, for example a nanoshell formed by superposing a sphere with a smaller void, will also show multiple resonances, hybrids of the individual structure modes.

The method can be extended to cover particles with size of the order of the wavelength by introducing a dynamic depolarization which varies across the particle \cite{Wokaun}, and extension to the full Mie theory permits a precise solution. However, although it is possible to find analytical solutions for simple structures, more complex structures frequently do not have a useful analytic solution. For this reason, numerical techniques are commonly used to predict plasmon resonances, as I will discuss in \fref{sec:numerical}.

\revisit{\begin{itemize}
	%\item \done{solutions: can proceed from a classical framework}
	%\item \done{solution of surface/symmetric film mode}
	\item \done{\TODO{solution of sphere - has resonances.}}
	\item \done{\TODO{Mie theory is needed for proper solutions when quasi-static limit not applicable}}
	%\item \done{multiple resonances}
	%\item \done{Only very simple geometries are analytically soluble $\Rightarrow$ FDTD}
\end{itemize}}







\section{Plasmon coupling}
\label{sec:coupling}
\Fref{fig:DispersionRelation} shows the dispersion relation found in \fref{eq:DispersionRelation} for plasmons at the interface of a dielectric with gold, described by experimental data from \cite{JohnsonAndChristy}. The confined plasmon modes of the interface are the section of the curve to the right of the light line. Above the transition frequency, the metal acquires dielectric character, and  radiation into the metal occurs.

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=0.9\textwidth]{DispersionRelation}%
	\caption{The blue curve shows the dispersion relation for plasmons at the interface of silver with air, using a silver dielectric function described by experimental data from \cite{JohnsonAndChristy}. The red curve shows the same dispersion relation using a Drude-model fit with negligible damping for the silver dielectric function. Sections of the dispersion curve to the right of the light line in air are propagating surface plasmons, while those to the left represent radiation into the metal. The right hand vertical scale is the same as the left hand frequency scale, but displays the equivalent free-space wavelength.}%
	\label{fig:DispersionRelation}%
\end{center}
\end{figure}

The larger $k$ vector of plasmon modes makes direct coupling to free-space radiation a highly suppressed process. For this reason, phase-matching schemes must be employed in order to efficiently generate plasmon polaritons, by introducing the extra momentum. A common method is prism coupling, as displayed in \fref{fig:CouplingSchemes}. In this configuration, light is totally internally reflected inside a prism. Light in the high-index dielectric has sufficiently large $k$ vector to excite plasmon polaritons at the air-metal interface of a thin film placed on the prism surface (in \fref{fig:DispersionRelation}, see that the light line in glass crosses the dispersion curve fro plasmons at the air-metal interface). This method requires that the film be sufficiently thin compared to the metal skin depth that the excitation can reach the air-metal interface. This coupling technique, is a form of attenuated total reflection (ATR), was originally proposed by Erwin Kretschmann in 1971\cite{Kretschmann1971}, and as a result is also termed the Kretschmann configuration. Another method for prism coupling, involves a prism brought close to a metal film on a separate substrate, such that the evanescent field produced by a totally internally reflected beam within the prism can excite surface plasmons at the film's air-metal interface. This method was originally proposed by Andreas Otto in 1968\cite{Otto}, and is termed the Otto configuration. The Kretschmann configuration is used more often, principally because it is simpler, not requiring the precise relative positioning of sample and prism. The two methods show a marked decrease in the intensity of the totally internally reflected light for incidence angles at which the component of the incoming light's $k$ vector matches that $k$ vector of the plasmon resonance with the same frequency.

\begin{figure}%
\begin{center}
	\leavevmode
	\includegraphics[width=120mm]{CouplingSchemes}%
	\caption{Plasmon coupling schemes. The rightmost diagram shows the Kretschmann configuration, whereby a totally internally reflected beam inside a prism has sufficient momentum to excite plasmons at the air-metal interface of a film on the prism surface, provided the film is sufficiently thin. The centre diagram shows the related Otto configuration, in which the evanescent field produced at a prism interface by a totally internally reflected beam couples to the air-metal surface plasmon mode of an interface placed sufficiently close to the prism edge. The rightmost diagram shows plasmon coupling via a structured surface. a periodic structure, such as a grating, can provide the necessary additional $k$ vector required to couple free-space radiation into the surface plasmon mode.}%
	\label{fig:CouplingSchemes}%
\end{center}
\end{figure}

Another, related method of providing the extra momentum required to couple free-space radiation into surface plasmon polariton modes is through the use of periodically structured materials, such as gratings or arrays of holes in a film. For example, a grating with base vector $\X{a}$ can contribute an additional pseudo-momentum $\X{q}=2\pi n\X{a}$. This technique also leads to increased coupling at specific angles of incidence, where the added pseudo-momentum makes up the difference between the $k$ vectors of the free-space radiation and the plasmon polariton mode of the same frequency. This effect occurs in the presence of any surface discontinuity, including isolated edges, but is much more effective when grating-type structures are employed.

Although a periodically-varying structure can be used effectively to add specific momentum vectors, in fact any variation in a structure can contribute extra momentum from scattering. This can alternatively be seen as a local relaxation of the dispersion relation due to hybridized modes. This is seen most frequently through the scattering which occurs at the edge of metal films or stripes. For example, when looking at a thin metal stripe waveguide, it is possible to excite plasmons in the waveguide mode simply by placing the exciting laser's focal spot on the end of the stripe. The discontinuity in the structure is enough to allow reasonable coupling between free-space and waveguide modes.

However, this ease of coupling can also present problems for applications of plasmonic structures. Such structures are typically fabricated using \EBL\ (see \fref{sec:EBL}) and thermal deposition of the metal of interest, most commonly gold or silver. Although the evaporation of metal onto the substrate to form a thin film does produce films with relatively well-controlled thickness, with uniform films down to a few \unit{}{nm} routinely produced, the process can still produce enough discontinuity to affect the coupling of plasmon modes in structures formed from such films. The film itself typically has a surface roughness with height variations in the region of \unit{\pm2}{nm} \cite{Kolomenski2009}, which is sufficient to cause increased losses for plasmon modes through re-radiation into the dielectric medium. Indeed, it is even possible to characterize a film's surface roughness by the alteration it makes to the plasmon polariton wave vector \cite{Hoffmann1998}. When viewed as an alteration to the plasmon wave vector, roughness increases both its real and imaginary components, thus leading both to a shorter plasmon wavelength and a decreased propagation length due to increased losses.

Furthermore, it is likely that in addition to surface roughness, local discontinuities in the metal's crystal structure in the form of grain boundaries also contribute to plasmon loss mechanisms. Thin metal films deposited by evaporation are necessarily grainy in composition, since the growth process is not atom by atom, but proceeds in clumps. In addition, the poor wetting properties of gold on dielectric substrates such as quartz lead it to preferentially form small particles rather than a smooth film. With metals on a longer length scale, it is possible to increase the size of grains, and hence the uniformity of the metal, through annealing, that is, heating the metal such that thermal excitations are sufficient to overcome the energy boundary which ordinarily prevents grains from restructuring into the lower-energy configuration of fewer, larger domains. However, for thin gold films, this is not possible, due again to its poor wetting properties. Gold does not adhere well to dielectric substrates such as quartz, and will tend to form small islands or droplets, akin to the water droplets observed on a strongly hydrophobic surface, albeit on a \unit{}{nm} scale rather than one of \unit{}{mm}. Indeed, in order to allow it to form distinct structures, a thin (of order \nm{1-5}) intermediate `adhesion' layer of another metal, typically chromium or titanium, is employed, to which the gold adheres more readily.

The local relaxation of dispersion relations at grain boundaries applies not only to the plasmon dispersion relation, but also to the electronic band structure of the metal itself. 

\Check{Discussion of fluorescence - I don't really know very much about why this occurs less in flakes than in thermal gold and also why it presents issues for the possibility of strong coupling}.

\revisit{
\begin{itemize}
	%\item \done{Dispersion relation}
	%\item \done{Kretschmann coupling}
	%\item \done{grating /lattice coupling}
	%\item \done{edge coupling}
	%\item \done{scattering from polycrystalline domains $\rightarrow$ extra loss.} \Check{How does this relate to fluorescence?}
	%\item \done{Loss in rough waveguides, \TODO{cite?}}
	%item dispersion relation strongly dependent on local dielectric environment
	%\item Large changes to structural response caused by minor irregularities in fabrication
	%\item problems with silver oxidation in air-exposed structures \TODO{cite?}
\end{itemize}}

%\section{Surface Acoustic Waves}
%\label{sec:SAW}
%A periodic surface structure can be induced by a \SAW\ travelling on a substrate. Surface acoustic waves can be induced and detected electrically using structures known as \IDTs. \TODO{add \IDT\ description}. Microwaves applied to the \IDT\ will generate surface acoustic waves if the frequency $\omega_m$ fulfils the relation $\omega_m=2\pi v_a /d_{IDT}$
%
%\begin{itemize}
%	\item piezoelectric media - non-linear effect. We earlier \Check{assumed linear responses} for our plasmonic structures!
%	\item different materials and cuts give different excitation coefficients and prop velocities
%	\item Rayleigh-type soliton (produced for particular crystal cuts)
%\end{itemize}

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